Construction of Bessel-Gauss Type Solutions for the Telegraph Equation

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Construction of Bessel-Gauss Type Solutions for the Telegraph Equation

Victor Borisov, Irina Simonenko

To cite this version:

Victor Borisov, Irina Simonenko. Construction of Bessel-Gauss Type Solutions for the Telegraph Equation. Journal de Physique I, EDP Sciences, 1997, 7 (8), pp.923-930. �10.1051/jp1:1997196�.

�jpa-00247376�

Construction of Bessel-Gauss Type Solutions for the Telegraph Equation

Victor V. Borisov

(*)

^{and Irina I. Simonenko}

Institute of Physics, St. Petersburg University, Ulyanovskaya 1, Petrodvorets, St.

Petersburg,

198904, Russia

(Received

8 March 1996, revised 19 November 1886, accepted it April

1997)

PACS.41.20.Jb Electromagnetic wave propagation; radiowave propagation

Abstract. The specific solutions of the telegraph equation ^for 3D _{space are} constructed.

These solutions enable _us to obtain the wavefunctions which _are akin to the Bessel-Gauss

wave

modes. The solutions of the homogeneous and

inhomogeneous

equations _are compared. The

application of the obtained _wave structures _to the description of the electromagnetic fields in _a

special case of inhomogeneity of the medium is discussed.

1. Introduction

In this paper we construct the

specific steady-state

solutions of the

homogeneous

and inho- mogeneous

telegraph

equations ^for 3D _{space as}

expansions

in terms of modes in

cylindric

coordinates. In _some

particular

_cases the obtained wavefunctions describe the solutions of the Klein-Gordon

equation

and of the

equation

for _waves in

lossy

media. Under certain conditions

on the

hyperplane

the

space-time

structure of the wavefunctions is akin to the structure of the Bessel-Gauss _wave modes obtained

by

Overfelt

iii

_as solutions of the

homogeneous

wave

equation.

In this _case the results _can be

applied

to the

description

of the localized electro-

magnetic

_waves in conductive medium and noncollisional

plasma including

_a

special

_case of

inhomogeneity

of the medium

along

_one dimension.

Investigation

of solutions to the inho- mogeneous equations is linked with the

possibility

of

generation

of localized _waves

by

_means

of _a

specific

_source,

moving

with the

velocity

^of

light

_[2], while the

original

localized modes have been obtained _as solutions of the

homogeneous

equation. Note that the

interpretation

of the latter solutions involves _some difficulties if _we

replace

the

inhomogeneous

medium

by homogeneous

free _space.

2. Solution of the

Homogeneous Equation

The

homogeneous telegraph equation

for 3D _space in

cylindric

coordinates _{p, q7, z} is

~°

⁺

a(fi))

₊

_fl(fi)(

₊

_{n(fi)) i~(P,}

v~,

fi, f2)

₌ o.

(i)

(* Author for correspondence (e-mail: borisfiphim.niif.spb.su, [email protected])

©

Les

#ditions

_{de Physique 1997}

924 JOURNAL DE PHYSIQUE I N°8

The notation D stands for

i7(

4

~~

=

~

p

^~

+

~~

4

~~

,

(i

2 ^{" T} ~ z,

~fif2

_P

~P ~P

_P

~i7 ~fi f2

'

where _T

= et is the time

variable,

_c is the wavefront

velocity.

We _assume that

a((i ), fl((i ),

and

n((i)

_are continuous functions of the variable

(I

We construct solution of the above equation with the

specific

condition _on

hyperplane

(1 " 0+

only [3,4]

il(P,

_v7, 0+,

f2)

=

F(P,

_v7)e~~~,

(2)

where k is constant. Then the function ~l may be

represented

in the

following

form i~(P>i7,

fi, f2)

" i~(P>_i7>fi)~~~~>

(~)

that

actually corresponds

to separation of the variable

(2.

Now from

equation (I)

and condi- tion

(2)

_{one can} ^{obtain for the function} 4l the second order

partial

differential equation:

~v[ ^{(4k a(ti ))} (

+

p(ti )k

₊

n(fi))

^{~ = o}

⁽⁴⁾

together

with the condition _on the

hyperplane

4~(P,v7,

fi

₌

0+)

=

F(P,

_v7).

(5)

With the

help

of the expansions

4~(P,_v~,

fi

=

~j

_4~m(P,

_{fi )e~'~~, F(P,}

v~) ⁼

~j

_Fm

_{(P)e~'~~, (6)}

m m

we separate ^the

angular

^variable _q?.

Using

^the Fourier-Bessel transform

Co Co

w(S>fl)

_"

/ dp pJm(Splw(p> fl), w(p>fl)

"

/

dS

SJm(Sp)w(S>fl),

O

where

Jm(sp)

is the Bessel function of the first kind of order _m and

W(p, (i)

is the function Fm _or

4lm,

_{one can} derive from

(4, 5)

^the equation _as well _as the

boundary

condition for the

coefficients 4lm

(s, (1)

(4k o(ti )) (~m (s, ti)

₊

(s2 fl(ti )k mini ))~m is, ti

₌ o,

4lm(S, fi

₌

0+)

=

Fm(S). (7)

Applying

the inverse transform _to the solution of the above

problem

4~m(s, ti)

₌

Fm(s)

_exp

j) _dti

^~~

(~~~~[~j ~~~~1, ₍₈₎

one obtains for the m-th coefficient of the

expansion (6)

4~m(P

fi)

= exP

Ill _df' ^~ll~~ _l~ill~ ^I"

_dS

SFmIS)Jm(PS)

f>

,

I

l~~~

x exP

-s~ /

+

~~i

4k

a(f'

Hence the

representation

of the wavefunction _{~l in terms} of modes in

cylindric

coordinates _may be written _as

i~(P>i7>

fl, f2)

_"

~

_i~m(P>i7,

_{fl, f2)}

"

Xl'lk~2

+

/~' df~ ~(~~~ ~~~~~

~ 0+ 1

If>

,

1

(10)

~

~

~im~

j"

ds

sfm(s)Jm(PS)

_~~~'

^~~~ +

~~~ ~~

°~G~

~ ~

This

expression gives

the

representation

of the wavefunction for different conditions _on the

hyperplane (1

_" 0+

(2).

3. Wave Structure of the Bessel-Gauss Mode

Type

We consider the solution of the

problem (1)

and

(2)

with the

particular

condition _on the

hyperplane

(1 " 0+

i~(P, v~,0+,f~)

₌ ^{~~~ ~~}f(v~)e~~~

@0

P

Here

d(p a)

is the Dirac distribution and _a is _a real _constant. We _assume that the func- tion

describing

the

angular

distribution of the wavefunction _on the

hyperplane

_may be rep- resented _as the finite _sum

f(q7)

₌

£$~~ fme~~~

The coefficients of

expression (6)

_are

Fm(p)

₌

)d(p ^{a) fm,}

m is

finite,

hence

Fm(s)

=

fmJm(as),

and

expression (8)

becomes

It,

,

p(t )k

₊

n(c)

~xp

-s2 /~' ^au

~~

~~~~j

Jm~~~~ _~~~~

~m(s, ti)

=

fm

_exP

/~

~~~

4k

a(fl)

°+

and

according

to

(9)

_we ^have

4~m(P,

fi

₌

fm

_exP

I/) dfl ~(j~~ ()~~~

x

/"

ds sJm

(Sal Jm(ps)

_exp ^-s2

/~' ^dtl

~~

_[~~, ⁽¹³⁾

o o+ 1

Using

the relation

(the

result 6.633.2 in [5])

/"

d~

~~-p2x2

_j

(~z)

j

(bz)

=

1

~-@

i ab

~14)

o

~ ~ 2p2 ^~ 2p2 ^'

where Re _{q >} -1, Re

p2

_> 0,

Iq(z)

is the modified Bes§el function of the first

kind,

and _{a, are} real constants, _{we may} rewrite the m-th term

~lm(p,

_q7,

(1, (2)

in the

following

form

i~m(P

_v7

fi f2)

=

lfm Ill dfi

4~

-l~n ⁱ

^Im

l~ Ill _df'

4~

-l~~~

j

x exp

k(2

+ lmi~ +

/)

~~~

~~~~~ ~~~~~

~~ ^~~

~~

^~~~_~~

^~~~

^~~~~

where Re

fl' _{d([ ~~_)}

~ > 0 and _m

(m

<

n)

is

integer.

926 JOURNAL DE PHYSIQUE I N°8

One _can check that the solution

(15)

^{satisfies the condition}

(11)

on the

hyperplane.

If

f f)( _d([

~~_)~~,

j

^~

» _m, with the

help

of the

expression Im(z)

cf

je~

_{we get}

«z

~~

_~~'

~'~~'

_~~~

~° ~i ^~

~~~ 4k

a(([

^~~~

If

(i

is small, the

integral fl' _d((

~~_)~~, ^cf

~(i,

^where _{Re ~} >

0,

then

using

the relation [6]

~

1 j2

~i /~

^~~~

~2E

^~~~~'

one obtains the condition _on the

hyperplane

(1 _" 0+

(11).

The

particular

_cases of the solution

(15)

are:

(I) ^The

telegraph equation

coefficients _a and

fl

_are real constants. Then from

(15)

_one obtains

i~m

(P

_v7

fi f2)

₌

fm ~li

^° Im

Ii _(4k

a))

x exP

imv~

+ ~~

~

~

flkfi

+

/~' ^{dfin(ti j}

+

kt~ ~~j

_P~

(4k a)

,

(16)

~+ i

where

Re(4k a)

_> 0. When ^~~

(4k al

» m, the above expression

gives 2(1

1

(4k oj

^~~~

l~m

(P'

_i~'

fl'

_~2

~'~

2/@

(1

~ ~~~

~~~

~ 4k _a

~~~~

~

~~

~~~~~~~

~ ~~~

~~~~~~

~~~ ~~

The factor

exp(k(2 ~f(4k _o))

_for

a = 0 describes the

specific

features of the localized

waves in free _space.

Nota~ly,

the space-time structure of the

single

term ~lm ^is akin to Bessel-

Gauss _{wave structure} described in

[ii.

(ii)

If in equation

ii)

_{o =}

fl

= 0, _we have the Klein-Gordon

equation. Using

the solution of the

problem

1°

+ ~

lfl ))~

" 0,

~(p>

v~> 0+,

f2

_"

F(p>

_v~)~~~~

Ii 7)

where

F(p,

_q7) is

f@ _f(q7),

one can obtain from

(15)

~m

₌

fm II

^Im

_(~i~k)

^exP

^imv7

⁺

£ II _dfinifi)

+ k

f2

^~~

i

^~~

₍₁₈₎

If

~fk(

» m is

satisfied,

~m

_m

Jm lexp _imvJ

+

( j) _dun(ti

+ k

t~

_~P

_~~~~

(iii)

We have the

homogeneous equition

for the

lossy

media in the

particular

_case where

a =

fl

₌ b and _{n =} 0 in equation

(1),

where b is _a

negative

constant. Then the wavefunction

~lm may be written _as

~lm =

fm

_exp

imq7

+

~~

(i

+

k(2

^{~~ ~ ~~}

(4k

_{b) x} ^~~ Im

~i _(4k ⁾⁾ ₍₁₉₎

4k b

4fi 2fi 2fi

4. Solution of the Bessel-Gauss Mode

Type

to the

Inhomogeneous Equation

We construct the solution of the

specific problem

D

+

aifi (

+

flifi )

+

nifi ))i~"'

⁼

l~ ^lip,

v~

fiie~~~

~l~" = 0, (1 < 0.

(20)

It is

supposed

that the _source is distributed _{on a}

cycle moving

with the constant

velocity

c

along

the axis _z. Then

)

=

e~~~ilP,

_v7,

fi)

=

~~~

j

^~~

f1v7)dlfi)e~~~,

121)

where

f(q7)

₌

£(~~ fme~'~~

One _can obtain the solution of the

problem (20) using

the

following representation

^of the wavefunction and the _source distribution

n n

~in

=

ekf2~jp,q~,ti)

=

~ ~t

= e~f2

~ 4~mip,ti)e~m~,

₁₂₂₁

m=o m=o

n n

j

₌

~j _jm

= ekf2

~ _{im(P, fi)e~~~ (23)}

m=o m=o

and

applying

the Fourier-Bessel transform. As _a result _we get the

following

equation

(4k o((i)))~m(s, ⁽¹⁾

⁺

^{(s2 p((i)k} n((i))~m(s, (i)

=

) fmJm(sa)&(ti),

with

boundary

condition:

4lm(S, 0-)

" 0.

Applying

the inverse transform to the solution of the above

problem

for the m-th term of

(22)

we find

~~ ~

~~~~~~

4k

a(0) ~'

~~~ 4k

-~(([) ^~~ i ^~'

_~~~

4k

-~(([

l

x exp

imv~

₊

_kt~

₊

i~' ^dfi ~ii~ _i~iii~

^P~

_1~~ [i~' ^au

_~~

^{~~~~ (24)}

Here

h((i)

is the Heaviside step ^{function and}

a(0)

is the value of the coefficient

a((i)

_on the

hyperplane

_{11 "} 0.

Comparing

the above result and the solution of the

homogeneous

equation

(15)

_{one can see} that ~lm =

fi(4k a(0))~lfl,

₍₁ > 0.

928 JOURNAL DE PHYSIQUE I N°8

5. Wavefunction of the Gauss Mode

Type

Using

the

expressions

of Section 2 _{one can} get the wavefunctions which _are akin to the Gauss

wave modes

provided

that

Fm(p)

₌

(p/dl'~e~P~/~~,

where _a and d _are constants. The substi- tution

Fm(s)

=

fis~e~%~

into

(10)

reduces to the representation

~2m+2

^f> ₁ ^'~

~~

~~'_~~

2~+1dm

~~~~~~~ ~~

~ ~

~

~~~ 4k

a(((

f>

~( (i

_~

fl

~i ~ ^f> 1

~~

~ ~~~

~

~~~

4( a(((j

^{~~~ ~~}

~~

~ ~

~

~~~ 4k

a(((

^~~~~

Inthecaseofo=fl=n=0wehave

~~ _~~'~~

~~~~~~

~'~~~~~

((ka)~~~i _l'~~~

^~~~

^fi~~~~£l)~

^'

that

corresponds

_to the result obtained

by Sezginer

_{[3] as} the localized solution of the homo- geneous wave

equation

in free _space. The

particular

_cases of solution

(25),

when _o,

fl,n

_are constant, are

given

ⁱⁿ

[7-10].

Ifthe condition _on the

hyperplane

(1 " 0+ is

~l(p, 0+, (2)

_"

e~f2d(p)/p,

that is

Fo(P)

_"

d(p)/p,

then from

(10)

we obtain

~° ~~'~~'~~~ ~~~~~

~

~~~ 4k

~(([

^~~~

~

~~~

~~ ~~~

x exP

I Ill df'

4~

-l~i~ j

(26)

Note that _{one can} obtain this result from

(15) taking

the limit _{a ~} 0.

If _a

=

fl

_{= n =}

0,

formula

(26)

^becomes

~lo(P,(1, (2)

"

~~fo

_exp

(2

^~ k

(27)

fi

~

Taking

into account the relation between the solutions _<of the

homogeneous

and

inhomogeneous

equations _{~lo "}

)k~l(n (a

₌

0),

_{one can see} that the above formula corresponds to the solution of the initial-value

problem

for the _waves in free _space due to the line current. Note that the

result

(27)

is the

particular

_case of the solution obtained in [11], ^which can be found

by

_means of the

algorithm

described in Section

2,

and with the

help

of the retarded solution of the _wave

equation.

6.

Application

of the Scalar Solutions to the

Descripti<on

of

Electromagnetic

Waves Some solutions of the

telegraph equation

obtained in the

previous

sections _can be

applied

to the

description

of the

electromagnetic

fields in different media. In the _case ofthe

homogeneous

Maxwell's

equations

and constant

conductivity

_{a one can} represent ^the components ^of the localized

electromagnetic

_waves in terms of two scalar

functions, using

the electric and

magnetic

one-component ^Hertz vectors n

=

ezll,

and n* ₌

e=II*

_[9]. Functions II and II* _are solutions of the

equation

^for the

lossy

media and the

single

term of the expression ^{for II} or II* is

given by expression (19),

where _c is the

velocity

of

light'and

=

-~]~. Only

TM solutions of the

inhomogeneous

Maxwell's

equations

in

conducting

media _can be described in _terms of the

scalar function II. Here the solution of the

telegraph equation

is the _sum ~l =

§f

+

~(~

^{II and}

the _source distribution

j

is the z-component of the external _current

density

vector

j

₌

ezjz.

In source-free _space, the localized TE

electromagnetic

_waves in noncollisional

plasma

_can be described with the

help

of the Hertz _vector n*

=

ezll*, provided

that the

plasma frequency

is constant [10]. ^Here one can obtain the scalar

potential

II* _as solution of

(18)

where

n((i)

is

constant.

It is worthwhile _to establish the relation between the obtained solutions of the scalar

telegraph equation

and the

description

of

electromagnetic

_waves in _an

inhomogeneous

medium. As _an

example,

_we discuss how to _use the obtained results for the

description

of the _waves in _an

inhomogeneous

conductor. In this _case the induced current

density

is

a((i )E,

the

conductivity

of medium is _a function of the variable

(i,

and E is the electric field

strength

vector.

One _can obtain from Maxwell's

equations

for the

magnetic

induction _vector B

"~~~~~

~~~j~~~)

₌

(ez~°

^x

j a((1)~~

az ~~

Hence for the

Bz

component we have the

equation

l~~ ~d/~(2 ^~~~~~ ~~l

^~

_~2~

^{~~ ~}

which is the

telegraph equation (1)

in which

a((i)

_"

fl((1)

"

-(4ir/c)a((i)

and

n((1)

" 0.

Using

the

expression (15)

we get ^the explicit solution for B=.

In the _case of the

axisymmetric

TE field _{one can} obtain for the

E~

component

l~P ^P~

^~

~~f2

~~~~

~l

~

~2 ^~~ ~l

~~~~

~~

~

Although

this equation is not the

telegraph equation,it

_can be solved in _a similar _way. Assum-

ing

that the condition _on the

hyperplane

(1 " 0+ is

E~(p,0+,(2)

_"

Fo(P)e~f2

^and

representing E~

_as in

(3) E~(p,(1,(2)

_"

E(p,(i) exp(k(2)

_{we are} faceded to the

following

equation:

~) )P

_14k

ajti))(

₊

pjti)k

+

njti)) ^{Ejp, ti)}

^{= o,}

with the

corresponding boundary

condition:

EIP, °+)

"

FolP),

Where

alfi)

₌

fllfi)

₌

~)alfi)

and

n(fi)

₌

-~j )a(fi).

Hence

by

_means of the Fourier- Bessel transform _{one can} obtain the

equivalent

to

(7),

where E

is, (1)

" Ah

Is, (i

but E

is, 0+)

=

Fo

Is).

If

Fo(p)

=

ii /p)d(p a)

_we find the solution for

E~ using

the

expansion (15)

ⁱⁿ which

Im(z)

=

fi(z)

and

exp(imq7)

₌ 1.

930 JOURNAL DE PHYSIQUE I N°8

7. Conclusion

In this _paper the solution

algorithm

is based _on the known method of

complete separation

of variables. We

consequently

separate the variables

(2

" T + z, q7, and p.

By

this _{means we} get

for the function of the variable

(i

" T z an

ordinary

differential equation of the first order.

Solving

this equation and

using

the inverse Fourier-Bessel transform _we obtain the solution of the

problem

under discussion. However the inverse Fourier-Bessel transform _can be

performed only

in

specific

_cases, this restricts the

possibility

of

constructing

_an

explicit

solution in the

space-time

representation. Notice that the

temporal

Fourier transform cannot be used here

because of the

special

structure of the solutions under considerations.

The

proposed

solution _can be used for the

description

of localized

electromagnetic

_waves

behind the ionization front

moving

with the

velocity

of

light.

While

propagating through

a

medium,

hard radiation

(X-rays)

interacts with its atoms

leading

to the

generation

of

charged particles (ions

and

electrons).

As _a result, we may consider the

plasma/neutral-gas boundary.

Because the front of the

X-rays pulse

"the ionization front" _moves with the

velocity

of

light,

the

boundary

has the _same

velocity.

The coefficients of the

telegraph

equation _a,

fl,

_{n are} to be taken constant when _we

neglect

the ionization

pulse

duration. The finite duration of the ionization

pulse

results in coefficients which _are functions of the variable

(i

_{" T z}

(see

_[12]

for _more

details).

Notice that

(i

is the observation time

computed

^from the time when the

X-rays

pulse ^front arrives at the observation

point (or,

_as

well,

the distance

computed

from the front of the ionization

pulse).

Acknowledgments

The research described in this _{paper was} made

possible

in part

by

^Grant M3H300 from the International Science Foundation and Russian Government and Russian Foundation for Basic Research Grant N° 96-02-17166.

References

[1] Overfelt P-L-,

Phys.

Rev. A 44

(1991)

3941.

[2] Borisov V.V. and Utkin

A-B-,

J. Math.

Phys.

35

(1994)

3624.

[3]

Sezginer A.,

J.

Appl. Phys.

57

(1985)

678.

[4] ^Hillion

P.,

J. Math.

Phys.

29

(1988)

1771.

[5]

Grandshteyn

I.S. and

Ryzhik I.M.,

Tables of

Integrals, Series,

and Products

(Academic Press,

New

York, 1969).

[6] Vladimirov

V.S., Equations

of Mathematical

Physics (Moscow, Nauka, 1988).

[7] Hillion P. J. Math.

Phys.

33

(1992)

1817.

[8]

Donelly

R. and Ziolkovski R.W. Proc. R. Sac. Land. A 437

(1992)

673.

[9] Borisov V-V- and Utkin

A.B.,

Can. J.

Phys.

72

(1994)

725.

[10] ^{Borisov V.V.} and Utkin

A.B.,

Can. J.

Phys.

72

(1994)

647.

[11] ^Borisov

V.V., Electromagnetic

Fields of Transient Currents

(St.-Petersburg

State Univer-

sity Press, St.-Petersburg, 1996).

[12] ^Borisov

V-V-, Nonsteady-State Electromagnetic

Waves

(Leningrad

State

University Press,

Leningrad, 1987).